几类二阶离散哈密顿系统同宿轨和异宿轨的存在性
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摘要
关于Hamilton系统同宿轨的研究可以追溯到Poincare关于天体力学的工作[51],Poincare发现如果系统的稳定流形和不稳定流形横截相交,那么这个系统就会有非常复杂的动力学行为并且含有无限多条同宿轨.之后,Birkhoff和Smale的工作深刻揭示了Poincare的发现中所蕴含的复杂动力学行为,也就是,一个具有横截同宿轨的系统是混沌的[62].因此,在非线性动力学的研究中,一个非常有意思的问题是在什么条件下我们可以证明一个系统具有横截同宿轨.著名的Melnikov方法对这个问题给出了部分的回答.一般说来,为了考察一个系统是否具有横截同宿轨,我们首先需要证明该系统中有同宿轨,其次再来讨论这个同宿轨的横截性.因此,研究一个系统的同宿轨或者异宿轨的存在性问题对于理解一个系统的复杂动力学行为是非常重要的一步.
自从Rabinowitz在文献[53]中使用变分方法研究一类一阶连续Hamilton系统的周期轨存在性问题的开创性工作之后,越来越多的研究者发现Hamil-ton系统具有某种变分结构,而这种变分结构在研究连续系统周期轨,同宿轨或者异宿轨的存在性问题上起到了非常重要的作用.而关于离散Hamilton系统相关问题研究开始不久,许多重要而有意思的问题尚待解决.
利用变分方法研究同宿轨或者异宿轨的存在性,我们通常需要在一个合适的Hilbert空间上面构造泛函,而这个泛函的非零临界点通常是我们所关心系统的同宿轨或者异宿轨.我们把寻找特殊轨道的问题转化成了寻找相应泛函非零临界点的问题,而变分方法在寻找临界点方面常常很有效.关于变分法方面的专著,读者可以参见文献[54,70].
连续Hamilton系统的同宿轨存在性的研究取得了许多重要进展Rabi-nowitz使用周期轨来逼近同宿轨的方法证明了一类二阶微分方程的同宿轨的存在性[56]Coti-Zelati和Rabinowitz讨论了一类在原点和无穷远点满足超二次条件的一类二阶Hamilton系统的同宿轨存在性问题[20].在文献[65]中,Sere引入多重碰撞解,并使用变分和Bernoulli转移的方法来刻画系统的复杂动力学行为.丁彦恒引入算子谱理论方面的工具研究了几类满足超二次或者次二次的二阶连续系统的同宿轨的存在性问题[27].
连续系统异宿轨的存在性研究也得到了好多重要结果.在文献[30]中,对一类满足一个空间变量是周期的另一个空间变量是超线性假设的方程,Felmer讨论了该系统异宿轨的存在性.在文献[9]中,Bertotti和Montecchiari对一类二阶几乎周期系统证明了有无限多个异宿轨连接两个退化的平衡点.在文献[13]中,Caldiroli和Jeanjean得到了存在一条异宿轨连接原点和一条极小的不可收缩周期轨的结论.在文献[55]中,Rabinowitz讨论了一类二阶Hamilton系统的周期和异宿轨的存在性问题Rabinowitz也得到了一些关于钟摆方程中异宿轨的存在性方面的结果[60]Coti-Zelati和Rabinowitz讨论了一类连接势能函数中的两个临界点的同宿轨的存在性问题,其中这两个临界点处在不同的能量层[21].
对Hamilton系统来说,尽管变分方法在寻找同宿轨方面非常有效,但如何证明该同宿轨的横截性仍然是一个困难的问题.因此,研究者引入“多重碰撞解”来研究Hamilton系统的复杂动力学行为.多重碰撞解的存在性问题,最早开始于Sere在一类一阶连续Hamilton系统的工作[651.之后,在很多其它微分系统中,多重碰撞解的存在性被证明.比如,一类满足退化条件的方程[59],阻尼系统[10],能量函数是变号的系统[14].关于差分方程方面的多重碰撞解的研究刚刚开始,关于差分方程同宿轨的多重碰撞解方面的研究非常少.
要证明多重碰撞解的存在性通常需要完成下面几步.首先,我们使用变分方法和极小极大方法找到一族非平凡的同宿轨,并且我们把这类同宿轨看作是“单次碰撞解”.然后,我们利用变分技巧来证明多重碰撞解的存在性,这类解在充分分开的几个时间段内与我们找到的单次碰撞解的距离非常近.在研究多重碰撞解的存在性中一个重要的假设是我们所考察泛函的临界点具有某种孤立性.这种孤立性的假设可以看作是比横截性条件稍微弱一些的假设.
对于一个实际系统进行建模和仿真对动力系统的研究起到了非常重要的作用.实际的计算是不能直接处理连续系统的,研究者通常都需要把连续系统转化成相应的差分系统以便通过仿真来观察系统的动力学行为.而且由于我们关心的问题不同,通常会给出不同形式的差分方程.另外,差分方程可以应用到物理,化学,工程等研究领域中去.因此,差分方程的研究逐渐变得重要起来.关于离散Hamilton系统的研究,可以参见文献[1].最近,许多关于离散Hamilton系统的周期轨,同宿轨和异宿轨存在性方面的重要结果被得到了[33,35,41,45,46,86,88].
最早利用变分方法研究差分方程的结果是文献[33]中的工作.郭志明和庾建设研究了如下的二阶纯量差分方程的周期解的存在性Δ2x(t-1)+f(t,x(t))=0.许多研究者讨论了如下类型的差分方程的周期轨,同宿轨和异宿轨的存在性的问题Δ(p(t)Δx(t-1))-L(t)x(t)=f(t,x(t)),x(t)∈Rn,t∈Z,这个方程可以通过一个适当的变换变成等价的离散Hamilton系统.在文献[46]中,马满军和郭志明在p(t),L(t)和f(t,x)关于时间变量t都是周期函数的假设下,研究了纯量差分方程同宿轨的存在性问题.在文献[45]中,在没有p(t)和L(t)是周期函数的假定下,假设f(t,x)在原点和无穷远处是超二次的,或者f(t,x)关于x是奇函数,马满军和郭志明得到差分方程存在非平凡的同宿轨的结论.在文献[41]中,林晓艳和唐先华在对所有的t∈Z,L(t)是正定矩阵且f(t,z)满足更一般的假设条件下证明差分方程有无限多条非平凡的同宿轨.
离散系统的异宿轨研究刚刚起步,有许多有趣的问题尚未解决.异宿轨的存在性是通过研究特定函数空间上的使得能量达到极小的元素而得到的.在文献[79]中,肖华峰和庾建设考察如下钟摆方程的异宿轨存在性问题：Δ2x(t-1)+a sin(x(t))=0,其中α∈R是参数,t∈Z,x(t)∈R.在文献[80,88]中,肖华峰等人以及张浩和李志祥研究如下差分方程的异宿轨的存在性问题：Δ2x(t-1)+V'x(x(t))=0,其中x(t)∈Rn,t∈Z.
本文主要讨论三个方面的问题：一是讨论了几类二阶离散哈密顿系统同宿轨的存在性问题;二是讨论了一类二阶离散哈密顿系统同宿轨的多重碰撞解;三是讨论了一类二阶离散哈密顿系统异宿轨的存在性问题.
针对上述三个问题,本文分为四章.第一章是准备知识,介绍变分方法的基本概念和方法以及线性算子谱理论的相关知识.
第二章主要研究了下面的二阶离散哈密顿系统Δ2x(t-1)-L(t)x(t)+V'x(t,x(t))-0, t∈Z,(*)其中Δx(t-1)=x(t)-x(t-1),Δ2x(t-1)=Δ(Δx(t-1))对任意t∈Z,L(t)是一个n×n实对称矩阵,V(t,·)∈C1(Rn,R)且Vx'(t,0)三0.我们的结果可以看作丁彦恒的结果的离散对应[27].之前关于差分方程同宿轨存在性的方面的工作通常假设对所有的t∈Z,L(t)是正定矩阵.在第二章中,我们利用差分算子谱理论来减弱这个假设.我们假设V(t,·)满足超二次或者次二次假设.我们不需要假设L和V是关于时间的周期函数,以及对所有的t∈Z,L(t)都是正定矩阵,我们证明了该差分方程至少存在一条非平凡的同宿轨.进一步,如果V(t,x)是超二次的并且关于x是偶函数,则该差分方程存在无限多条非平凡的同宿轨.为了说明所得到结果,我们给出了两个具体例子.
在第三章中,我们研究二阶离散哈密顿系统(*).我们假设V(t,·)是变号函数,L和V是周期函数.我们首先证明同宿轨的存在性.然后,假设临界点满足孤立性条件,我们研究了多重碰撞解的存在性.在差分方程方面,多重碰撞解的研究刚刚开始.据我们所知,我们还没有见到相关文献讨论差分方程同宿轨的多重碰撞解的问题.我们的结果可以看作Caldiroli和Montecchiari的结果的离散对应[14].
在第四章中,我们研究下面一类二阶离散哈密顿系统异宿轨的存在性问题：Δ2x(t-1)-μL(t)x(t)+W'x(t,x(t),δ)=0,t∈Z,其中W(t,x,δ)=a(t)V(x,δ),x∈Rn;Δx(t-1)=x(t)-x(t-1),Δ2x(t-1)=Δ(Δx(t-1))；对任意t∈Z,L(t)是正定矩阵；a(·)：Z→R是周期函数；V(·,δ)∈C2(Rn,R4),并且V(x,·)是连续的；μ∈[0,1]和δ∈[0,δ0]是参数且δ0>0.在一定的假设条件下,我们得到若δ和μ充分小,则存在一条异宿轨连接函数V(x,δ)的两个具有不同临界值的临界点.当μ=0时,我们的结果可以看作文献[21]中结果的离散对应.
据我们所知,现有关于讨论差分方程异宿轨存在性的文献都是假设μ=0并且该异宿轨连接的是函数V的两个具有相同值的临界点,也就是那两个临界点具有相同的势能[88,79,80].而对于连续系统的异宿轨研究方面,我们还没有见到相关文献研究μ≠0的情况.
The research of homoclinic orbits of Hamiltonian systems dates back to Poincare's work on celestial mechanics [51], in which Poincare discovered that the dynamical behavior of the system is very complicated and there exist infinitely many homoclinic orbits if the stable and unstable manifolds intersect transversally. Later, the work of Smale and Birkhoff showed that a system with a transversal homoclinic orbit is chaotic [62]. It is a meaningful question to determine whether or not a system has a transversal homoclinic orbit. The well known Melnikov method answers a part of this problem. Generally speaking, to investigate the existence of transversal homoclinic orbit, we need to show the existence of the homoclinic orbit and show this orbit is transversal. Hence, it is a very important step to show the existence of homoclinic or heteroclinic orbits.
After the pioneering work of Rabinowitz [53], studying the existence of periodic solutions of a class of first order continuous Hamiltonian systems by the variational methods, more and more researchers found that the variational methods play an important role in the investigation of the existence of periodic, homoclinic and heteroclinic orbits of continuous Hamiltonian systems because of the variational structure. The research on the discrete Hamiltonian system is just beginning, there are lots of unsolved problems in this area.
To investigate the existence of homoclinic or heteroclnic orbits via the variational method, one usually needs to construct a suitable variational func-tional on a proper Hilbert space such that a non-zero critical point of the functional is a non-trivial homoclinic or heteroclinic orbit of the system. So, we convert this problem to the study of the existence of the non-zero critical points of the functional. And, the variational approach is very effective in find-ing the critical points. We refer to [54,70] for more materials on variational methods.
A great progress has been made in the research of the existence of ho-moclinic orbits of Hamiltonian systems. In [56], Rabinowitz obtained the ex-istence of a homoclinic orbit of a kind of second-order differential equations by using a series of periodic orbits to approximate the homoclinic orbit. In [20], Coti-Zelati and Rabinowitz investigated the existence of homoclinic or-bits of second-order Hamiltonian systems with the superquadratic potential assumptions at both the origin and at the infinity. In [65], Sere introduced the multibump solutions and depicted the complicated dynamical behavior combined with the Bernoulli shift. In [27], Ding applied the spectral theo-ry to investigate several types of supquadratic or subquadratic second-order differential equations.
There are lots of good results that have been obtained in the study of the existence of heteroclinic orbits. In [30], Felmer investigated the existence of heteroclinic orbits of a kind of one-order Hamiltonian systems with the as-sumption that the Hamiltonian is periodic with respect to one space variable and superlinear in another one. In [9], Bertotti and Montecchiari studied the existence of infinitely many heteroclinic solutions connecting degenerate equi-libria for a kind of second-order almost periodic systems. In [13], Caldiroli and Jeanjean obtained that there is a heteroclinic orbit connecting the origin and a minimal non-contractible periodic orbit, which is the limit of a sequence of homoclinic orbits with special properties. In [55], Rabinowitz studied the exis-tence of periodic and heteroclinic orbits of a class of second-order Hamiltonian systems. There are also some works on the existence of heteroclinic orbits for the pendulum equation [60]. Coti-Zelati and Rabinowitz investigated the ex-istence of heteroclinic orbits which connect two critical points of the potential function at different energy levels [21].
For Hamiltonian system, although the variational method is very powerful, it is difficult to show the transversality of the homoclinic orbit. A kind of relatively weaker solution "multibump solution" is introduced to study the complicated dynamics of Hamiltonian systems. For the multibump solutions of certain systems, it is Sere's who firstly investigated the multibump solutions of a kind of first-order Hamiltonian systems under certain conditions [65]. Later, similar construction of the orbits is obtained in different situations. For example, the degenerate case [59], the damped systems [10], the potential changing sign case [14]. However, the construction of multibump solutions in difference equations is just beginning, there is few work about the existence of multibump solutions of homoclinic orbits of difference equations.
The procedure to show the existence of multibump solutions is to first use a variational argument, minimax method, to find a special family of non-trivial homoclinic solutions which can be regarded as the "one-bump" solution-s. Secondly, variational arguments are applied to get multibump solutions, i.e. solutions near sums of sufficiently separated translates of the "one-bump" so-lutions. A key assumption for the existence of multibump solutions is that the critical points of the corresponding functional are isolated. This hypothesis is used to replace the classical transversality conditions.
For the research of dynamical systems, the method of modeling and sim-ulating of a real system plays an important role. Since the continuous systems can not be directly applied in the real computation, we need to transform the continuous equations into their corresponding difference equations such that we could observe the dynamical behavior of the systems through simulation-s. Hence, the research of difference equations gradually becomes important. Discrete Hamiltonian systems can be applied in many different areas, such as physics, chemistry, engineering and so on. Please refer to [1] for more infor-mation on discrete Hamiltonian systems. There are many good results on the existence of periodic, homoclinic, and heteroclinic orbits of discrete Hamilto-nian systems [33,35,41,45,46,86,88].
In [33], Guo and Yu investigated the existence of periodic and subhar-monic solutions of the scalar second-order difference equation as follows: Δ2x(t-1)+f(t,x(t))=0. Lots of researchers have studied the existence of periodic, homoclinic, and heteroclinic orbits of the following type of difference equations Δ(p(t)Ax(t-1))-L(t)x(t)=f(t,x(t)), x(t)∈Rn, t∈Z, which can be written as equivalent discrete Hamiltonian systems through a proper transformation. In [46], Ma and Guo proved the existence of homoclin-ic orbits of difference equations with n=1under the periodicity assumptions on p(t), L(t), and f(t,x) in t. In [45], Ma and Guo studied the existence of homoclinic orbits of difference equation provided that f(t,x) grows superlin-early both at origin and at infinity or f(t,x) is an odd function with respect to x, where the assumption of periodicity on p(t) and L(t) are not required. In [41], Lin and Tang obtained that there exist infinitely many homoclinic orbits of the equation with the assumption that L(t) is positive definite for any t∈Z and more general conditions on f(t,z).
The research of the existence of heteroclinic orbits of discrete systems is just beginning. The existence of heteroclinic orbits can be shown by studying the existence of minimizing sequences on certain function spaces. In [79], Xiao and Yu investigated the existence of heteroclinic orbits of the following discrete pendulum equation A2x(t-1)+a sin(x(t))=0, where a∈R is a parameter, x(t)∈R, and t∈Z. In [80,88], Xiao et al. and Zhang and Li studied the existence of heteroclinic orbits of the following difference equation Δ2x(t-1)+V'x(x(t))=0, where x(t)∈Rn and t∈Z.
This thesis deals with three basic problems:the first is to investigate the existence of homoclinic orbits of two types of second-order discrete Hamiltonian systems; the second is to study the existence of multibump solutions of a class of second-order discrete Hamiltonian systems; the third is to look for the heteroclinic orbits of a kind of second-order discrete Hamiltonian systems.
In Chapter1, some basic concepts and useful results about variational methods are introduced, and some knowledge on the spectral theory of linear operators is given.
In Chapter2, we discuss the following type of second-order discrete Hamil-tonian systems: Δ2x(t-1)-L(t)x(t)+V'x(t,x(t))＝0,t∈Z,(*) where Δx(t-1)=x(t)-x(t-1), Δ2x(t-1)=Δ(Δx(t-1)), L(t) is an n×n real symmetric matrix for each t∈Z, and V(t,·)∈C1(Rn, R) for each t∈Z with V'x(t,0)=0. Our results can be regarded as a discrete analog of Ding's results obtained in [27]. In the previous work about the existence of homoclinic orbits of difference equations, the assumption that L(t) is positive definite for each t∈Z was required. In the present paper, we try to weaken this assumption by the spectral theory of difference operators. Assume that V(t,·) is superquadratic or subquadratic. We do not assume that L and V are periodic functions, and L(t) is positive definite for all t∈Z. We show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t,x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end of this chapter, two illustrative examples are provided.
In Chapter3, we investigate the existence of multibump solutions of the second-order discrete Hamiltonian systems in the form of (*). We assume that V(t,·) is a sign-changing function, L and V are periodic functions. First, we show the existence of homoclinic orbit. Then, we discuss the multibump solu-tions with the assumption on the isolation of critical points by the variational methods. The study of the multibump solutions of the difference equations is just starting. As we know, there are very few results on the research of multibump solutions of homoclinic orbits of difference equations. Our results can be regarded as a discrete analog of Caldiroli and Montecchiari's results obtained in [14].
In Chapter4, we study the existence of the heteroclinic orbits of the following type of second-order discrete Hamiltonian systems: Δ2x(t-1)-μL(t)x(t)+W'x(t, x(t),δ)=0, t∈Z, where W(t,x,δ)=a(t)V(x,δ), x∈Rn, L(t) is a positive definite matrix for any t∈Z, a(·):Z→R is a periodic function, V(·,δ)∈C2(Rn,R), and V(x,·) is continuous, and μ∈[0,1] and δ∈[0, δ0] are parameters with δ0>0. Under certain conditions, we show that for sufficiently small δ and μ, there exists at least one heteroclinic orbit connecting two critical points of the function V(x,δ). When μ=0, our results can be regarded as a discrete analog of Coti-Zelati and Rabinowitz's results obtained in [21].
As we know, the previous works on the existence of heteroclinic orbits of difference equations are about the case that μ=0and the heteroclinic orbits connecting two critical points of V with the same value, that is, the two critical points are at the same energy level [88,79,80]. For the work on the existence of heteroclinic orbits of continuous systems, there is no result about the case μ≠0.
自从Rabinowitz在文献[53]中使用变分方法研究一类一阶连续Hamilton系统的周期轨存在性问题的开创性工作之后,越来越多的研究者发现Hamil-ton系统具有某种变分结构,而这种变分结构在研究连续系统周期轨,同宿轨或者异宿轨的存在性问题上起到了非常重要的作用.而关于离散Hamilton系统相关问题研究开始不久,许多重要而有意思的问题尚待解决.
利用变分方法研究同宿轨或者异宿轨的存在性,我们通常需要在一个合适的Hilbert空间上面构造泛函,而这个泛函的非零临界点通常是我们所关心系统的同宿轨或者异宿轨.我们把寻找特殊轨道的问题转化成了寻找相应泛函非零临界点的问题,而变分方法在寻找临界点方面常常很有效.关于变分法方面的专著,读者可以参见文献[54,70].
连续Hamilton系统的同宿轨存在性的研究取得了许多重要进展Rabi-nowitz使用周期轨来逼近同宿轨的方法证明了一类二阶微分方程的同宿轨的存在性[56]Coti-Zelati和Rabinowitz讨论了一类在原点和无穷远点满足超二次条件的一类二阶Hamilton系统的同宿轨存在性问题[20].在文献[65]中,Sere引入多重碰撞解,并使用变分和Bernoulli转移的方法来刻画系统的复杂动力学行为.丁彦恒引入算子谱理论方面的工具研究了几类满足超二次或者次二次的二阶连续系统的同宿轨的存在性问题[27].
连续系统异宿轨的存在性研究也得到了好多重要结果.在文献[30]中,对一类满足一个空间变量是周期的另一个空间变量是超线性假设的方程,Felmer讨论了该系统异宿轨的存在性.在文献[9]中,Bertotti和Montecchiari对一类二阶几乎周期系统证明了有无限多个异宿轨连接两个退化的平衡点.在文献[13]中,Caldiroli和Jeanjean得到了存在一条异宿轨连接原点和一条极小的不可收缩周期轨的结论.在文献[55]中,Rabinowitz讨论了一类二阶Hamilton系统的周期和异宿轨的存在性问题Rabinowitz也得到了一些关于钟摆方程中异宿轨的存在性方面的结果[60]Coti-Zelati和Rabinowitz讨论了一类连接势能函数中的两个临界点的同宿轨的存在性问题,其中这两个临界点处在不同的能量层[21].
对Hamilton系统来说,尽管变分方法在寻找同宿轨方面非常有效,但如何证明该同宿轨的横截性仍然是一个困难的问题.因此,研究者引入“多重碰撞解”来研究Hamilton系统的复杂动力学行为.多重碰撞解的存在性问题,最早开始于Sere在一类一阶连续Hamilton系统的工作[651.之后,在很多其它微分系统中,多重碰撞解的存在性被证明.比如,一类满足退化条件的方程[59],阻尼系统[10],能量函数是变号的系统[14].关于差分方程方面的多重碰撞解的研究刚刚开始,关于差分方程同宿轨的多重碰撞解方面的研究非常少.
要证明多重碰撞解的存在性通常需要完成下面几步.首先,我们使用变分方法和极小极大方法找到一族非平凡的同宿轨,并且我们把这类同宿轨看作是“单次碰撞解”.然后,我们利用变分技巧来证明多重碰撞解的存在性,这类解在充分分开的几个时间段内与我们找到的单次碰撞解的距离非常近.在研究多重碰撞解的存在性中一个重要的假设是我们所考察泛函的临界点具有某种孤立性.这种孤立性的假设可以看作是比横截性条件稍微弱一些的假设.
对于一个实际系统进行建模和仿真对动力系统的研究起到了非常重要的作用.实际的计算是不能直接处理连续系统的,研究者通常都需要把连续系统转化成相应的差分系统以便通过仿真来观察系统的动力学行为.而且由于我们关心的问题不同,通常会给出不同形式的差分方程.另外,差分方程可以应用到物理,化学,工程等研究领域中去.因此,差分方程的研究逐渐变得重要起来.关于离散Hamilton系统的研究,可以参见文献[1].最近,许多关于离散Hamilton系统的周期轨,同宿轨和异宿轨存在性方面的重要结果被得到了[33,35,41,45,46,86,88].
最早利用变分方法研究差分方程的结果是文献[33]中的工作.郭志明和庾建设研究了如下的二阶纯量差分方程的周期解的存在性Δ2x(t-1)+f(t,x(t))=0.许多研究者讨论了如下类型的差分方程的周期轨,同宿轨和异宿轨的存在性的问题Δ(p(t)Δx(t-1))-L(t)x(t)=f(t,x(t)),x(t)∈Rn,t∈Z,这个方程可以通过一个适当的变换变成等价的离散Hamilton系统.在文献[46]中,马满军和郭志明在p(t),L(t)和f(t,x)关于时间变量t都是周期函数的假设下,研究了纯量差分方程同宿轨的存在性问题.在文献[45]中,在没有p(t)和L(t)是周期函数的假定下,假设f(t,x)在原点和无穷远处是超二次的,或者f(t,x)关于x是奇函数,马满军和郭志明得到差分方程存在非平凡的同宿轨的结论.在文献[41]中,林晓艳和唐先华在对所有的t∈Z,L(t)是正定矩阵且f(t,z)满足更一般的假设条件下证明差分方程有无限多条非平凡的同宿轨.
离散系统的异宿轨研究刚刚起步,有许多有趣的问题尚未解决.异宿轨的存在性是通过研究特定函数空间上的使得能量达到极小的元素而得到的.在文献[79]中,肖华峰和庾建设考察如下钟摆方程的异宿轨存在性问题：Δ2x(t-1)+a sin(x(t))=0,其中α∈R是参数,t∈Z,x(t)∈R.在文献[80,88]中,肖华峰等人以及张浩和李志祥研究如下差分方程的异宿轨的存在性问题：Δ2x(t-1)+V'x(x(t))=0,其中x(t)∈Rn,t∈Z.
本文主要讨论三个方面的问题：一是讨论了几类二阶离散哈密顿系统同宿轨的存在性问题;二是讨论了一类二阶离散哈密顿系统同宿轨的多重碰撞解;三是讨论了一类二阶离散哈密顿系统异宿轨的存在性问题.
针对上述三个问题,本文分为四章.第一章是准备知识,介绍变分方法的基本概念和方法以及线性算子谱理论的相关知识.
第二章主要研究了下面的二阶离散哈密顿系统Δ2x(t-1)-L(t)x(t)+V'x(t,x(t))-0, t∈Z,(*)其中Δx(t-1)=x(t)-x(t-1),Δ2x(t-1)=Δ(Δx(t-1))对任意t∈Z,L(t)是一个n×n实对称矩阵,V(t,·)∈C1(Rn,R)且Vx'(t,0)三0.我们的结果可以看作丁彦恒的结果的离散对应[27].之前关于差分方程同宿轨存在性的方面的工作通常假设对所有的t∈Z,L(t)是正定矩阵.在第二章中,我们利用差分算子谱理论来减弱这个假设.我们假设V(t,·)满足超二次或者次二次假设.我们不需要假设L和V是关于时间的周期函数,以及对所有的t∈Z,L(t)都是正定矩阵,我们证明了该差分方程至少存在一条非平凡的同宿轨.进一步,如果V(t,x)是超二次的并且关于x是偶函数,则该差分方程存在无限多条非平凡的同宿轨.为了说明所得到结果,我们给出了两个具体例子.
在第三章中,我们研究二阶离散哈密顿系统(*).我们假设V(t,·)是变号函数,L和V是周期函数.我们首先证明同宿轨的存在性.然后,假设临界点满足孤立性条件,我们研究了多重碰撞解的存在性.在差分方程方面,多重碰撞解的研究刚刚开始.据我们所知,我们还没有见到相关文献讨论差分方程同宿轨的多重碰撞解的问题.我们的结果可以看作Caldiroli和Montecchiari的结果的离散对应[14].
在第四章中,我们研究下面一类二阶离散哈密顿系统异宿轨的存在性问题：Δ2x(t-1)-μL(t)x(t)+W'x(t,x(t),δ)=0,t∈Z,其中W(t,x,δ)=a(t)V(x,δ),x∈Rn;Δx(t-1)=x(t)-x(t-1),Δ2x(t-1)=Δ(Δx(t-1))；对任意t∈Z,L(t)是正定矩阵；a(·)：Z→R是周期函数；V(·,δ)∈C2(Rn,R4),并且V(x,·)是连续的；μ∈[0,1]和δ∈[0,δ0]是参数且δ0>0.在一定的假设条件下,我们得到若δ和μ充分小,则存在一条异宿轨连接函数V(x,δ)的两个具有不同临界值的临界点.当μ=0时,我们的结果可以看作文献[21]中结果的离散对应.
据我们所知,现有关于讨论差分方程异宿轨存在性的文献都是假设μ=0并且该异宿轨连接的是函数V的两个具有相同值的临界点,也就是那两个临界点具有相同的势能[88,79,80].而对于连续系统的异宿轨研究方面,我们还没有见到相关文献研究μ≠0的情况.
The research of homoclinic orbits of Hamiltonian systems dates back to Poincare's work on celestial mechanics [51], in which Poincare discovered that the dynamical behavior of the system is very complicated and there exist infinitely many homoclinic orbits if the stable and unstable manifolds intersect transversally. Later, the work of Smale and Birkhoff showed that a system with a transversal homoclinic orbit is chaotic [62]. It is a meaningful question to determine whether or not a system has a transversal homoclinic orbit. The well known Melnikov method answers a part of this problem. Generally speaking, to investigate the existence of transversal homoclinic orbit, we need to show the existence of the homoclinic orbit and show this orbit is transversal. Hence, it is a very important step to show the existence of homoclinic or heteroclinic orbits.
After the pioneering work of Rabinowitz [53], studying the existence of periodic solutions of a class of first order continuous Hamiltonian systems by the variational methods, more and more researchers found that the variational methods play an important role in the investigation of the existence of periodic, homoclinic and heteroclinic orbits of continuous Hamiltonian systems because of the variational structure. The research on the discrete Hamiltonian system is just beginning, there are lots of unsolved problems in this area.
To investigate the existence of homoclinic or heteroclnic orbits via the variational method, one usually needs to construct a suitable variational func-tional on a proper Hilbert space such that a non-zero critical point of the functional is a non-trivial homoclinic or heteroclinic orbit of the system. So, we convert this problem to the study of the existence of the non-zero critical points of the functional. And, the variational approach is very effective in find-ing the critical points. We refer to [54,70] for more materials on variational methods.
A great progress has been made in the research of the existence of ho-moclinic orbits of Hamiltonian systems. In [56], Rabinowitz obtained the ex-istence of a homoclinic orbit of a kind of second-order differential equations by using a series of periodic orbits to approximate the homoclinic orbit. In [20], Coti-Zelati and Rabinowitz investigated the existence of homoclinic or-bits of second-order Hamiltonian systems with the superquadratic potential assumptions at both the origin and at the infinity. In [65], Sere introduced the multibump solutions and depicted the complicated dynamical behavior combined with the Bernoulli shift. In [27], Ding applied the spectral theo-ry to investigate several types of supquadratic or subquadratic second-order differential equations.
There are lots of good results that have been obtained in the study of the existence of heteroclinic orbits. In [30], Felmer investigated the existence of heteroclinic orbits of a kind of one-order Hamiltonian systems with the as-sumption that the Hamiltonian is periodic with respect to one space variable and superlinear in another one. In [9], Bertotti and Montecchiari studied the existence of infinitely many heteroclinic solutions connecting degenerate equi-libria for a kind of second-order almost periodic systems. In [13], Caldiroli and Jeanjean obtained that there is a heteroclinic orbit connecting the origin and a minimal non-contractible periodic orbit, which is the limit of a sequence of homoclinic orbits with special properties. In [55], Rabinowitz studied the exis-tence of periodic and heteroclinic orbits of a class of second-order Hamiltonian systems. There are also some works on the existence of heteroclinic orbits for the pendulum equation [60]. Coti-Zelati and Rabinowitz investigated the ex-istence of heteroclinic orbits which connect two critical points of the potential function at different energy levels [21].
For Hamiltonian system, although the variational method is very powerful, it is difficult to show the transversality of the homoclinic orbit. A kind of relatively weaker solution "multibump solution" is introduced to study the complicated dynamics of Hamiltonian systems. For the multibump solutions of certain systems, it is Sere's who firstly investigated the multibump solutions of a kind of first-order Hamiltonian systems under certain conditions [65]. Later, similar construction of the orbits is obtained in different situations. For example, the degenerate case [59], the damped systems [10], the potential changing sign case [14]. However, the construction of multibump solutions in difference equations is just beginning, there is few work about the existence of multibump solutions of homoclinic orbits of difference equations.
The procedure to show the existence of multibump solutions is to first use a variational argument, minimax method, to find a special family of non-trivial homoclinic solutions which can be regarded as the "one-bump" solution-s. Secondly, variational arguments are applied to get multibump solutions, i.e. solutions near sums of sufficiently separated translates of the "one-bump" so-lutions. A key assumption for the existence of multibump solutions is that the critical points of the corresponding functional are isolated. This hypothesis is used to replace the classical transversality conditions.
For the research of dynamical systems, the method of modeling and sim-ulating of a real system plays an important role. Since the continuous systems can not be directly applied in the real computation, we need to transform the continuous equations into their corresponding difference equations such that we could observe the dynamical behavior of the systems through simulation-s. Hence, the research of difference equations gradually becomes important. Discrete Hamiltonian systems can be applied in many different areas, such as physics, chemistry, engineering and so on. Please refer to [1] for more infor-mation on discrete Hamiltonian systems. There are many good results on the existence of periodic, homoclinic, and heteroclinic orbits of discrete Hamilto-nian systems [33,35,41,45,46,86,88].
In [33], Guo and Yu investigated the existence of periodic and subhar-monic solutions of the scalar second-order difference equation as follows: Δ2x(t-1)+f(t,x(t))=0. Lots of researchers have studied the existence of periodic, homoclinic, and heteroclinic orbits of the following type of difference equations Δ(p(t)Ax(t-1))-L(t)x(t)=f(t,x(t)), x(t)∈Rn, t∈Z, which can be written as equivalent discrete Hamiltonian systems through a proper transformation. In [46], Ma and Guo proved the existence of homoclin-ic orbits of difference equations with n=1under the periodicity assumptions on p(t), L(t), and f(t,x) in t. In [45], Ma and Guo studied the existence of homoclinic orbits of difference equation provided that f(t,x) grows superlin-early both at origin and at infinity or f(t,x) is an odd function with respect to x, where the assumption of periodicity on p(t) and L(t) are not required. In [41], Lin and Tang obtained that there exist infinitely many homoclinic orbits of the equation with the assumption that L(t) is positive definite for any t∈Z and more general conditions on f(t,z).
The research of the existence of heteroclinic orbits of discrete systems is just beginning. The existence of heteroclinic orbits can be shown by studying the existence of minimizing sequences on certain function spaces. In [79], Xiao and Yu investigated the existence of heteroclinic orbits of the following discrete pendulum equation A2x(t-1)+a sin(x(t))=0, where a∈R is a parameter, x(t)∈R, and t∈Z. In [80,88], Xiao et al. and Zhang and Li studied the existence of heteroclinic orbits of the following difference equation Δ2x(t-1)+V'x(x(t))=0, where x(t)∈Rn and t∈Z.
This thesis deals with three basic problems:the first is to investigate the existence of homoclinic orbits of two types of second-order discrete Hamiltonian systems; the second is to study the existence of multibump solutions of a class of second-order discrete Hamiltonian systems; the third is to look for the heteroclinic orbits of a kind of second-order discrete Hamiltonian systems.
In Chapter1, some basic concepts and useful results about variational methods are introduced, and some knowledge on the spectral theory of linear operators is given.
In Chapter2, we discuss the following type of second-order discrete Hamil-tonian systems: Δ2x(t-1)-L(t)x(t)+V'x(t,x(t))＝0,t∈Z,(*) where Δx(t-1)=x(t)-x(t-1), Δ2x(t-1)=Δ(Δx(t-1)), L(t) is an n×n real symmetric matrix for each t∈Z, and V(t,·)∈C1(Rn, R) for each t∈Z with V'x(t,0)=0. Our results can be regarded as a discrete analog of Ding's results obtained in [27]. In the previous work about the existence of homoclinic orbits of difference equations, the assumption that L(t) is positive definite for each t∈Z was required. In the present paper, we try to weaken this assumption by the spectral theory of difference operators. Assume that V(t,·) is superquadratic or subquadratic. We do not assume that L and V are periodic functions, and L(t) is positive definite for all t∈Z. We show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t,x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end of this chapter, two illustrative examples are provided.
In Chapter3, we investigate the existence of multibump solutions of the second-order discrete Hamiltonian systems in the form of (*). We assume that V(t,·) is a sign-changing function, L and V are periodic functions. First, we show the existence of homoclinic orbit. Then, we discuss the multibump solu-tions with the assumption on the isolation of critical points by the variational methods. The study of the multibump solutions of the difference equations is just starting. As we know, there are very few results on the research of multibump solutions of homoclinic orbits of difference equations. Our results can be regarded as a discrete analog of Caldiroli and Montecchiari's results obtained in [14].
In Chapter4, we study the existence of the heteroclinic orbits of the following type of second-order discrete Hamiltonian systems: Δ2x(t-1)-μL(t)x(t)+W'x(t, x(t),δ)=0, t∈Z, where W(t,x,δ)=a(t)V(x,δ), x∈Rn, L(t) is a positive definite matrix for any t∈Z, a(·):Z→R is a periodic function, V(·,δ)∈C2(Rn,R), and V(x,·) is continuous, and μ∈[0,1] and δ∈[0, δ0] are parameters with δ0>0. Under certain conditions, we show that for sufficiently small δ and μ, there exists at least one heteroclinic orbit connecting two critical points of the function V(x,δ). When μ=0, our results can be regarded as a discrete analog of Coti-Zelati and Rabinowitz's results obtained in [21].
As we know, the previous works on the existence of heteroclinic orbits of difference equations are about the case that μ=0and the heteroclinic orbits connecting two critical points of V with the same value, that is, the two critical points are at the same energy level [88,79,80]. For the work on the existence of heteroclinic orbits of continuous systems, there is no result about the case μ≠0.
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